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# The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer

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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
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Bunuel wrote:
The sequence S is defined by $$S_n = S_{n - 1} + S{n - 2} - 1$$ for each integer n ≥ 3. If $$S_1 = 11$$ and $$S_3 = 10$$, what is the value of $$S_5$$?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19

Bunuel Could you please format the equation properly? It's currently confusing. Thanks!

$$S_n = S_{n - 1} + S{n - 2} - 1$$ —> $$S_n = S_{n - 1} + S_{n - 2} - 1$$
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
dabaobao wrote:
Bunuel wrote:
The sequence S is defined by $$S_n = S_{n - 1} + S{n - 2} - 1$$ for each integer n ≥ 3. If $$S_1 = 11$$ and $$S_3 = 10$$, what is the value of $$S_5$$?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19

Bunuel Could you please format the equation properly? It's currently confusing. Thanks!

$$S_n = S_{n - 1} + S{n - 2} - 1$$ —> $$S_n = S_{n - 1} + S_{n - 2} - 1$$

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Edited. Thank you.
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
Strategy: Utilize a Table for the sequence to help understand how the sequence works

Given
s1 = 11
s3 = 10 = s2 + s1 - 1

Question
s5 = ?

Equations used
sn = s(n-1) + s(n-2) + 1

Calculations
s5 = s4 + s3 - 1
s5 = s4 + 11 - 1
What is s4?

s4 = s3 + s2 - 1
s4 = 11 + s2 - 1
To find s4, we need to know what s2 is

From the Given:
s3 = s2 + s1 - 1
s3 - s1 + 1 = s2
10 - 11 + 1 = s2
0 = s2

s4 = s3 + s2 -1
s4 = 10 + 0 - 1
s4 = 9

s5 = s4 + s3 - 1
s5 = 9 + 10 - 1
s5 = 18
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
Bunuel wrote:
The sequence S is defined by $$S_n = S_{n - 1} + S_{n - 2} - 1$$ for each integer n ≥ 3. If $$S_1 = 11$$ and $$S_3 = 10$$, what is the value of $$S_5$$?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19

Steps:
1) Compute S_2 = S_3 - S_1 + 1 = 0
2) Using the same method, compute S_4 = 10 - 1 = 9
3)Now, compute S_5 = 9 + 10 - 1 = 18

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The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
Good night Bunuel !

I am trying to understand this problem but still do not get it, why are we assuming that S1 + S2 will be equal to S3?

Does this mean that every integer after the last one is going to be -1 from the sum of the las two? I think my main problem is that I am not understanding the wording well enough.

$$S_n = S_{n - 1} + S_{n - 2} - 1$$

Could someone help me, please?

Kind regards!
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
jfranciscocuencag wrote:
Good night Bunuel !

I am trying to understand this problem but still do not get it, why are we assuming that S1 + S2 will be equal to S3?

Does this mean that every integer after the last one is going to be -1 from the sum of the las two? I think my main problem is that I am not understanding the wording well enough.

$$S_n = S_{n - 1} + S_{n - 2} - 1$$

Could someone help me, please?

Kind regards!

We are told that the sequence S is defined by $$S_n = S_{n - 1} + S_{n - 2} - 1$$ for each integer n ≥ 3.

This means that $$S_n = S_{n - 1} + S_{n - 2} - 1$$ is a formula which gives you every term of the sequence starting from 3rd. Look at the formula, it says that $$S_n$$ equals to the sum of two preceding terms, $$S_{n - 1}$$ and $$S_{n - 2}$$, minus 1. For example:

$$S_3 = S_{3 - 1} + S_{3 - 2} - 1=S_{2} + S_{1} - 1$$;
$$S_4 = S_{4 - 1} + S_{4 - 2} - 1=S_{3} + S_{2} - 1$$;
$$S_5 = S_{5 - 1} + S_{5 - 2} - 1=S_{4} + S_{3} - 1$$;
...

For more check the links below.

12. Sequences

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ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
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